ABSTRACT. It has often been remarked that the metatheory of strong reduction >-, the
combinatory analogue of βη-reduction -->>βη in λ-calculus, is rather complicated. In
particular, although the confluence of >- is an easy consequence of -->>βη being confluent,
no direct proof of this fact is known. Curry and Hindley’s problem, dating back
to 1958, asks for a self-contained proof of the confluence of >-, one which makes no
detour through λ-calculus. We answer positively to this question, by extending and
exploiting the technique of transitivity elimination for ‘analytic’ combinatory proof
systems, which has been introduced in previous papers of ours. Indeed, a very short
proof of the confluence of >- immediately follows from the main result of the present
paper, namely that a certain analytic proof system G_e[C], which is equivalent to
the standard proof system CL_ext of Combinatory Logic with extensionality, admits
effective transitivity elimination. In turn, the proof of transitivity elimination — which,
by the way, we are able to provide not only for G_e[C] but also, in full generality, for
arbitrary analytic combinatory systems with extensionality —employs purely proof theoretical
techniques, and is entirely contained within the theory of combinators.

ABSTRACT. It has often been remarked that the metatheory of strong reduction >-, the
combinatory analogue of βη-reduction -->>βη in λ-calculus, is rather complicated. In
particular, although the confluence of >- is an easy consequence of -->>βη being confluent,
no direct proof of this fact is known. Curry and Hindley’s problem, dating back
to 1958, asks for a self-contained proof of the confluence of >-, one which makes no
detour through λ-calculus. We answer positively to this question, by extending and
exploiting the technique of transitivity elimination for ‘analytic’ combinatory proof
systems, which has been introduced in previous papers of ours. Indeed, a very short
proof of the confluence of >- immediately follows from the main result of the present
paper, namely that a certain analytic proof system G_e[C], which is equivalent to
the standard proof system CL_ext of Combinatory Logic with extensionality, admits
effective transitivity elimination. In turn, the proof of transitivity elimination — which,
by the way, we are able to provide not only for G_e[C] but also, in full generality, for
arbitrary analytic combinatory systems with extensionality —employs purely proof theoretical
techniques, and is entirely contained within the theory of combinators.